Two results on the Convex Algebraic Geometry of sets with continuous symmetries
Renato G. Bettiol, Mario Kummer, Ricardo A. E. Mendes
TL;DR
This work analyzes convex sets invariant under continuous group actions. It first constructs a constructive, $\\mathsf{G}$-equivariant spectrahedral description for $\\mathsf{G}$-invariant spectrahedra by lifting to a $\\mathsf{G}$-representation and then restricting to a finite-dimensional subspace. It then uses Kostant's convexity for polar representations to transfer invariant convex geometry between a representation space $V$ and a section $\\mathfrak{a}$, proving that convex semi-algebraic sets, spectrahedral shadows, and rigidly convex sets are preserved under the induced bijection (with conjectures about broader preservation, such as spectrahedra). These results yield symmetry-reduction techniques for convex optimization, reducing dimensionality without enlarging the class of solvable problems under SDP, and clarify what invariants survive the polar correspondence. Altogether, the paper advances practical SDP reformulations in symmetric settings while detailing fundamental limitations and open questions in convex algebraic geometry under symmetry.
Abstract
We prove two results on convex subsets of Euclidean spaces invariant under an orthogonal group action. First, we show that invariant spectrahedra admit an equivariant spectrahedral description, i.e., can be described by an equivariant linear matrix inequality. Second, we show that the bijection induced by Kostant's Convexity Theorem between convex subsets invariant under a polar representation and convex subsets of a section invariant under the Weyl group preserves the classes of convex semi-algebraic sets, spectrahedral shadows, and rigidly convex sets.